Autoionization: Recent Developments and Applications

PHYSICS OF ATOMS AND MOLECULES Series Editors
P. G. Burke, The Queen's University of Belfast, Northern Ireland H. K1einpoppen, Atomic Physics Laboratory, University of Stirling, Scotland
Editorial Advisory Board
R. B. Bernstein (New York, U.S.A.) J. C. Cohen-Tannoudji (Paris, France) B. W. Crompton (Canberra, Australia) J. N. Dodd (Dunedin, New Zealand) G. F. Drukarev (Leningrad, U.S.S.R.) W. Hanle (Giessen, Germany)
C. J. Joachain (Brussels, Belgium) W. E. Lamb, Jr. (Tucson, U.S.A.) P.-O. LOwdin (Gainesville, U.S.A.) H. O. Lutz (Bielefeld, Germany) M. R. C. McDowell (London, U.K.) K. Takayanag! (Tokyo, Japan)
ATOM-MOLECULE COLLISION THEORY: A Guide for the Experimentalist Edited by Richard B. Bernstein
ATOMIC INNER-SHELL PHYSICS Edited by Bernd Crasemann
ATOMS IN ASTROPHYSICS Edited by P. G. Burke, W. B. Eissner, D. G. Hummer, and I. C. Percival
AUTOIONIZATION: Recent Developments and Applications Edited by Aaron Temkin
COHERENCE AND CORRELATION IN ATOMIC COLLISIONS Edited b~ H. K1einpoppen and J. F. Williams
DENSITY MATRIX THEORY AND APPLICATIONS Karl Blum
ELECTRON AND PHOTON INTERACTIONS WITH ATOMS Edited by H. K1einpoppen and M. R. C. McDowell
ELECTRON-ATOM AND ELECTRON-MOLECULE COLLISIONS Edited by Juergen Hinze
ELECTRON-MOLECULE COLLISIONS Edited by Isao Shimamura and Kazuo Takayanagi
INNER-SHELL AND X-RAY PHYSICS OF ATOMS AND SOLIDS Edited by Derek J. Fabian, Hans K1einpoppen, and Lewis M. Watson
INTRODUCTION TO THE THEORY OF LASER-ATOM INTERACTIONS Marvin H. Mittleman
ISOTOPE SHIFTS IN ATOMIC SPECTRA W. H. King
PROGRESS IN ATOMIC SPECTROSCOPY, Parts A, B, and C Edited by W. Hanle, H. K1einpoppen, and H. J. Beyer
VARIATIONAL METHODS IN ELECTRON-ATOM SCATTERING THEORY R. K. Nesbet
A Continuation Order Plan is available for this series. A continuation order will bring delivery of each new volume immediately upon publication. Volumes are billed only upon actual shipment. For .further information please contact the publisher.
AUTOIONIZATION Recent Developments
National Aeronautics and Space Administration Goddard Space Flight Center
Greenbelt, Maryland
Library of Congress Cataloging in Publication Data
Main entry under title:
Autoionization: recent developments and applications.
(Physics of atoms and molecules) Includes bibliographies and index. 1. Auger effect. I. Temkin, Aaron. II. Series.
QC793.5.E627A96 1985 539.7'2112
ISBN -13: 978-1-4684-4879-5 e-ISBN -13: 978-1-4684-4877-1 DOl: 10.1007/978-1-4684-4877-1
© 1985 Plenum Press, New York Softcover reprint of the hardcover 1 st edition 1985
A Division of Plenum Publishing Corporation 233 Spring Street, New York, N.Y. 10013
All rights reserved
85-6334
No part of this book may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, microfilming,
recording, or otherwise, without written permission from the Publisher
CONTRIBUTORS
A. K. BHATIA Atomic Physics Office Laboratory for Astrophysical and Solar Physics Goddard Space Flight Center National Aeronautics and Space Administration Greenbelt. M D
KWONG T. CHUNG Department of Physics North Carolina State University Raleigh. NC
BRIAN F. DAVIS Department of Physics North Carolina State University Raleigh. NC
GEORGE A. DoSCHEK Eo 00 Hurlburt Center for Space Research Naval Research Laboratory Washington. DoC.
B. R. JUNKER Office of Naval Research Arlington. VA
C. WILLIAM MCCURDY Department of Chemistry Ohio State University Columbus. OH
A. TEMKIN Atomic Physics Office Laboratory for Astronomy and Solar Physics Goddard Space Flight Center National Aeronautics and Space Administration Greenbelt. MD
v
PREFACE
About five years ago, Professor P. G. Burke asked me to edit a sequel to an earlier book-Autoionization: Theoretical, Astrophysical, and Laboratory Experimental Aspects, edited by A. Temkin, Mono Book Corp., Baltimore, 1966. Because so much time had gone by and so much work had been done, the prospect of updating the 1966 volume seemed out of the question.
In 1965 the phenomenon of autoionization, although long known, was just starting to emerge from a comparatively intuitive stage of understanding. Three major developments characterized that development: In solar (astro-)physics, Alan Burgess (1960) had provided the resolution of the discrepancy of the temperature of the solar corona as observed versus that deduced from ionization balance calculations, by including the process of dielectronic recombination in the calculation; Madden and Codling (1963) had just performed their classic experiment revealing spectroscopically sharp lines in the midst of the photoionization continuum of the noble gases; and Feshbach (1962) had developed a theory with the explicit introduction of projection operators, which for the first time put the calculation of auto ionization states on a firm theoretical footing. There were important additional contributions made at that time as well; nevertheless, without going into further detail, we were able to include in our 1966 volume, in spite of its modest size, a not too incomplete survey of the important developments at that time.
To do the equivalent now would be virtually impossible. In considering the alternatives, I felt that laboratory experimental developments in particular have far outstripped what can reasonably be included in the confines of a single book. Therefore, I have omitted them completely. The situation with regard to solar and astrophysical applications at first seemed also too vast for inclusion. However, the unlikely has become fact by virtue of a magnificent effort by Dr. George Doschek. His chapter, "Diagnostics of Solar and Astrophysical Plasmas Dependent on Autoionization Phenomena," is, in my opinion, a masterful exposition and summary of diagnostic analysis and applications of autoionization in almost the entire realm of space physics. It is necessarily a large part of this book. I hope the reader will find it enlightening and useful. It will surely have a vital place in the space physics literature.
The remaining chapters I have chosen to include are devoted to theory and calculation. Even here a severe limitation was required, but in the belief that good theory allows good calculations, and the value of calculations cannot exceed the quality of their theoretical underpinnings, we could be selective.
vii
Vlll PREFACE
In the category oftheory-calculation we could certainly have included an overview of methods and programs, developing mainly from the close coupling formalism, that dealt directly with electron scattering including resonances. Fortunately, there have been a number of recent reviews, so we have not felt it mandatory to include such a review here. Too recent to be included here and in a somewhat different category are successful develop ments, primarily calculational in nature, including resonances in many-body diagrammatic and random phase approximation (RPA) techniques. In contrast, there has been very little written of a review nature on the calculation of electron-atom (atomic ion) resonances within the context of the Feshbach theory. Since that theory has long provided the theoretical basis for much of the work of the Goddard group, I believe the present volume is a very appropriate place to present such a review. In our first article, Dr. Bhatia and I have attempted to review, from a more pedagogical point of view rather than from one of completeness, our work on two-electron systems (one-electron targets) for which theory allows explicit and rigorous projection operators to be given. We have included, however, a more detailed exposition of a recent calculation ofthe line-shape parameter, because that requires a rather different approach to a part of the Feshbach theory known as the nonresonant continuum. I believe the idea of a more generally defined nonresonant continuum may be of value in other contexts as well.
In a second article, Dr. Bhatia and I have undertaken the process of implementing the Feshbach approach to more than two-electron systems. As a prerequisite for actually doing calculations, we have found it necessary to precisely define the projection operators (P and Q) in complete and explicit terms. We have chosen to include a part of that analysis here because it also serves the pedagogical aims we have also attempted to fulfill. In the second part ofthat chapter we have discussed approximations ofthese operators that we have called quasi-projection operators (I' and ~). Historically our introduction of these quasi-projection operators preceded our recent develop ment of the projection operators themselves. Notwithstanding, quasi projections allow for meaningful calculations to be done, and we have briefly reviewed some of them.
The fact that one can calculate with projectors without them being idempotent (the latter property usually being implicit in the definition of the name "projection operator") is not confined to the specific quasi-projection operators we have introduced. In the third chapter of this book Drs. K. T. Chung and Brian K. Davis describe a hole-projection formalism wherein electrons in inner orbitals are projected out of an otherwise general ansatz for the wave function of the total system by a projection-type operator, which can certainly be considered in the category of quasi projectors. Their theory relies on a mini-max theorem which (although rigorously proved only in a one-
PREFACE IX
electron context) states that the physically meaningful state is realized when the energy is minimized with respect to the parameters of the complete wave function, but at the same time the energy is maximized with respect to the parameters describing the excluded orbitals (the holes). It is clear that the formalism should be particularly effective in calculating inner-shell vacancies of many-electron systems; however, even for 3-electron systems, to which the calculations have thus far been confined, as the article will show, the results are very impressive. In their summation the authors refer to a recent paper wherein they have combined their hole-projection method with elements of complex rotation to calculate widths. I expect that augmentation to become an important addition to the methodology.
Complex rotation is the subject of the last set of articles in this volume. The basic idea can be expressed in many ways, but for the purposes of this Preface one way is to notice that a stationary (i.e., bound)-state wave function has the time dependence exp ( - iEt/h), where E is a real number. Therefore, if a state is decaying, it should be describable by a complex time dependence W = E - ir/2; then its imaginary part will automatically describe the decay width (the inverse of the decay time) of the resonance. From the calculational point of view this has the implication that certain states, which are not quadratically integrable on the real axis, do become integrable off the real axis. This (measure zero) set of discrete states are uncovered-according to a basic theorem of Balslev and Combes-if the electronic coordinates are rotated in the complex plane beyond a minimum amount which, not surprisingly, is related to the width of the resonance. From the calculational point of view, however, this is a most important fact, because it removes boundary conditions from the problem. Thus, it implies that one can, in principle, calculate a many-electron resonant state without knowing the wave function of the target system. This, in turn (and again, in principle), overcomes a major shortcoming of the projection-operator approach, wherein although the eigenfunctions of QHQ are discrete and exist on the real axis, the projection operator Q does depend on the eigenfunctions of the target system and therefore must be approximated for more than one-electron target systems. In addition, both shape and Feshbach resonances can emerge from the complex rotation approach.
Briefly stated, Dr. B. R. Junker concentrates on applications to atoms and ions, whereas Dr. C. M. McCurdy deals with molecular systems; both authors have made a concerted effort to coordinate their respective treat ments. An important element of the approach of these two articles is the idea that it is preferable to retain the Hamiltonian in its real form and put the complex nature of the calculation completely in the ansatz for the wave function. How best to do this is not yet completely settled, but I believe these treatments go a long way in elucidating the technique. For simpler systems
x PREFACE
(e.g., He -) the results are probably the most reliably accurate of any thus far obtained. We are pleased to have these two contributions from two expert practitioners whose interests are calculational as well as theoretical.
I would like to thank all the authors for their contributions, and Professors P. G. Burke and H. Kleinpoppen for their encouragement. I am as usual indebted to Dr. A. K. Bhatia, in this case for his additional help in preparing the index.
Silver Spring. Maryland AARON TEMKIN
CONTENTS
RESONANCES AND AUTOIONIZA TION OF TWO-ELECTRON ATOMS AND IONS
A. TEMKIN AND A. K. BHATIA
1. Introduction . . . . . . . . . . . . . . . . . . . . 1 2. The Feshbach Formalism. . . . . . . . . . . . . . . . 2 3. The Projection-Operator Formalism of the Line-Shape Parameter q 13 4. Relationship of Breit-Wigner and Feshbach Resonance Parameters. 15 5. Variational Calculations of Cn • • . • • • • • . 17 6. Precision Calculation of the Lowest 1 S Resonance in
Electron-Hydrogen Scattering. . . . . . . . . 21 7. Review of Semiprecision Calculations of Other Fundamental
Autoionization States of He and H-. . . . . . . . . 28
CHAPTER TWO PROJECTION AND QUASI-PROJECTION OPERATOaS
FOR ELECTRON IMPACT RESONANCES ON MANY-ELECTRON ATOMIC TARGETS
1. Introduction . . . 2. Idempotent Resonance Projection Operators .
2.1. Notation and Preliminaries . . . . . 2.2. Derivation of Idempotent Projection Operator 2.3. Realization of P for the One-Electron Target . 2.4. Determination of P for the Two-Electron Target.
3. Quasi-Projection Operators. . . . 3.1. Justification and Representations 3.2. Spurious States. . . . . . . 3.3. Previous Calculations . . . . 3.4. He-ePO): A New Type of Feshbach Resonance
4. Conclusions . . . . . . . . . . . . 4.1. Resume and Future Work. . . . . . 4.2. Calculation of Nonresonant Phase Shifts.
xi
35 37 37 38 44 47 51 52 55 62 63 69 6~
70
FESHBACH RESONANCES AND INNER-SHELL VACANCIES
KWONG T. CHUNG AND BRIAN F. DAVIS
1. Introduction . . 2. Holes-Projection Theorem . . . . . . 3. Excited Bound States-A Test Calculation 4. Resonances of Helium below the n = 2, n = 3, and n = 4 Thresholds of He +
5. He- Resonances with a Singly or Doubly Excited Core 5.1. Singly Excited Core. . . . . . . . . . . 5.2. Doubly Excited Core . . . . . . . . . .
6. Li Resonances with a Singly or Doubly Excited Core. 7. Summary ............. .
CHAPTER FOUR COMPLEX STABILIZATION METHOD
B. R. JUNKER
2.1. Spectrum of the Rotated Hamiltonian. 2.2. Gamow-Siegert Boundary Conditions
3. Previous Calculations. . . . . . . . 3.1. Analytical Models and Analyticity of the S Matrix 3.2. Numerical Complex-Coordinate Calculations
4. The Complex-Stabilization Method . . . . . . . 5. The Variational Principle. . . . . . . . . . . 6. Structure of Variational Wave Functions and Basis Sets.
6.1. Basis Functions . . . . . . . . . . . . . 6.2. Potential Scattering. . . . . . . . . . . . 6.3. Electron-Atom and Electron-Molecule Resonances 6.4. Field Ionization . . .
7. Model Potential Calculations 8. Discussion. . . . . . .
CHAPTER FIVE MOLECULAR RESONANCE CALCULATIONS: APPLICATIONS OF COMPLEX-COORDINATE
AND COMPLEX BASIS FUNCTION TECHNIQUES C. WILLIAM MCCURDY
1. Introduction . . . . 2. Complex Coordinates and Complex Basis Functions: The Essential Ideas.
73 74 78 80 88 88 93 95 99
103 105 105 106 109 109 114 116 120 121 121 122 122 123 125 131
135 138
CONTENTS xiii
2.1. Complex Coordinates . . . . . 138 2.2. Complex Basis Functions. . . . 141 2.3. The Complex Variational Principle. 142
3. Features Peculiar to the Molecular Problem . 143 3.1. Branch Points in the Nuclear Attraction Potentials. 143 3.2. Practical Basis Set Methods and the Details of Why They Work 146 3.3 Numerical Stability and the Calculation of Molecular Photoionization
Cross Sections. . . . . . . . . . . . . . . . . .. 148 3.4. Summary of the Complex Basis Function Approach for Molecules 150
4. The Complex Self-Consistent-Field (CSCF) Method . . . . . .. 151 4.1. The Working Equations . . . . . . . . . . . . . .. 151 4.2. Continued Fraction Procedure for Finding the Stationary Point of ECSCF 154
5. CSCF Calculations on Molecular Shape Resonances 155 5.1. The Lowest 2r.: shape Resonance States of Hi 155 5.2. The 2ITg Shape Resonance State of Ni . . . 158
6. The Question of Correlation in Shape Resonances. 162 7. Questions Still Unanswered. . . . . . . . . 164
7.1. Extensions and Limitations of the Complex Basis Function Method. 164 7.2. Using the Complex Variational Principle. without Complex
Basis Function Calculations. . . . . . . . . . . . . . . 166
GEORGE A. DOSCHEK
1. Introduction . . . . . 2. Solar and Astrophysical Plasmas
2.1. The Solar Atmosphere. . 2.2. Solar Flares. . . . . . 2.3. Stellar Atmospheres and Other Astrophysical Sources .
3. Plasma Diagnostics and Autoionization. . 3.1. Theory of Spectral Line Formation. . 3.2. Electron Temperature Measurements . 3.3. Electron Density Measurements. 3.4. Ionization Equilibrium .
INDEX. . . . . . . . . .
171 174 174 180 182 183 183 186 210 245
257
AUTOIONIZATION OF TWO ELECTRON ATOMS AND IONS
1. INTRODUCTION
In this chapter, we shall review and discuss the theory and calculation of autoionization states of two-electron systems from the point of view of projection operator formalism as used by the authors over many years. Autoionization states are resonances caused by scattering an electron impinging on a target ion or atom. They can also be produced by photoabsorption as well as heavy-particle collisions. It is only comparatively recently that such resonances have been studied in great detail as a result of important experimental and theoretical developments together with the advent of high-speed computers.
Here, we shall confine ourselves to theoretical and calculational develop ments emanating from the Feshbach formalism and its application to scattering electrons from one-electron (hydrogenlike) targets. Such re sonances are by definition autoionization states of the composite-in this case, two-electron-system. One of the purposes of studying such systems is the hope of being able to achieve accuracy approaching what has been attained for ordinary two-electron bound states. To that extent we are testing quantitatively the continuum solutions of the Schrodinger equation to an accuracy far beyond what has previously been forthcoming from ordinary (i.e., nonresonant) scattering.
In Section 2, we briefly review the Feshbach formalism(l) with emphasis on how it applies to one-electron targets. Of particular interest is the fact that
A. TEMKIN and A. K. BHATIA • Atomic Physics Office, Laboratory for Astronomy and Solar Physics, Goddard Space Flight Center, National Aeronautics and Space Administration, Greenbelt, MD 20771.
2 A. TEMKIN AND A. K. BHATIA
the resonance parameters that emerge from the Feshbach approach are not identical to the corresponding Breit - Wigner quantities. Furthermore, we demonstrate how the two are related. Special attention will be devoted to the Fano line shape parameter(2) q (Section 3) and how it can be unambiguously formulated in projection-operator form.
These theoretical developments will be complemented by a review of our calculations for IS, 3p, and 1 D states of H- below the first excited threshold of the target H atom and by comparison with other accurate calculations and experiments. Similar calculations of 1,3S, 1,3 P, and 1,3D states for He, which is the second basic two-electron system we deal with, are made in Sections 4, 5, and 6.
2. THE FESHBACH FORMALISM
In this section, we describe briefly the Feshbach(1) theory for two-electron systems. Given a fixed hydrogen like target and a second electron, the Schrodinger equation is written in the usual way
(2.1)
Here, we take 'P(r1,r2) to be an eigenfunction of definite angular momentum, spin, and parity. Spin is included by requiring the spatial 'P to be antisym metric with respect to the exchange of the r1 and r2 coordinates for triplet states, and symmetric for singlet states; H is the total Hamiltonian of the system
1i2 Ze2 Ze2 e2 H= --(V~+V~)----+- (2.2)
2m r1 r2 r12
Z is the total charge of the nucleus, and E in Eq. (2.1) is the total energy. (In general, we shall use rydberg units Ii = 1, m = 1/2, e2 = 2.)
The essence ofthe Feshbach formalism(1) is the introduction of projection operators P and Q such that
P+Q=1 completeness (2.3 a)
P2=P} idem potency (2.3 b)
Q2=Q (2.3 c)
PQ= 0 orthogonality (2.3 d)
Equations (2.3b) and (2.3c) are the conditions that operators P and Q are projection operators (i.e., idempotent). From Eq. (2.1), we obtain by
TWO-ELECTRON ATOMS AND IONS
P(H - E)(P + Q)'P = 0
Q(H - E)(P + Q)'P = 0
And from Eq. (2.5), we obtain by "division" the formal expression
1 Q'P = Q(E _ H)Q QHP'P
Substituting Eq. (2.6) into Eq. (2.4) results in
p[ H + HQ Q(E ~ H)Q QH - E ]p'P = 0
3
(2.4)
(2.5)
(2.6)
(2.7)
The significance of the projection operators P and Q in the Feshbach theory comes from the condition that the P'P have the same asymptotic form as 'P
lim P'P = 'P (2.8)
Equation (2.8) implies that as far as the scattering is concerned, the same information is obtained from P'P as 'P. Equation (2.3a) leads to the asymptotic condition on Q'P
lim Q'P-+O (2.9) rl orr2-+OO
This is the same condition that is obtained for bound states. Thus, we have divided the total wave function 'P into scatteringlike and bound-state-like components. The significance of this is, as we shall see, that Eq. (2.7) is essentially a one-body equation in which all the many-body effects have been condensed into a one-body optical potential
1 1'oP = PHQ E _ QHQ QHP (2. lOa)
However, as useful as Eq. (2.7) and the optical potentials are, they are not a panacea; 1'oP' for example, is nonlocal, and to construct it exactly would be as difficult as solving the original SchrOdinger equation.
Let us first consider a specific form of projection operators (the preceding conditions do not specify P and Q uniquely) for systems containing two identical particles (electrons). It has been shown by Hahn, O'Malley, and Spruch(3) in a basic paper introducing the Feshbach formalism into the atomic-scattering problem (it was originally developed in the context of nuclear physics) that for two-electron systems, projection operators P and
Q satisfying the requirements in Eqs. (2.8) and (2.9) are given by
Q = Q1Q2
(2.11 )
(2.12a)
(2.12b)
Since from Eq. (2.12b) Pl = Pi' we have from Eqs. (2.3 a) and (2.12a) the explicit expressions for P and Q
Q = (1- Pd(1- P2) = 1- P1 - P2 + P1P2
P=I-Q=P1 +P2-P1P2
(2.13a)
(2.13b)
It may readily be verified that P and Q are idempotent and orthogonal (P2 = P, Q2 = Q; PQ = 0). It is useful to remember that P acting on the wave function 'P projects onto the ground state of the target and Q projects out of the ground state. In Eq. (2.12b), cPo is the ground state of the hydrogen like target
(2.14)
The major implication of the optical potential can be gleaned by expanding it into a complete set of eigenfunctions of the projected Hamil tonian QHQ, i.e.,
QHQn(r 1, r 2 ) = Cn<l>n(r 1, r 2)
Since Q2 = Q
Q2 HQn(r 1,r2) = QHQn(r1,r2 ) = CnQn(r1,r2 )
(2.15)
This implies that the eigenfunctions <l>n are also eigenfunctions of the operator Q, with the eigenvalues equal to 1 (we shall describe this by saying the <l>n are in Q space). Inserting the completeness of <l>n in Q space
into Eq. (2.lOa), we obtain
"Y = I,PHQn> < <l>nQHP op n E - Cn
(2. lOb)
Throughout this chapter "Yop as expanded in Eq. (2.10b) will playa central role. First and foremost for our purposes here is the fact that it can predict
TWO-ELECTRON ATOMS AND IONS 5
resonances. This is heuristically obvious from Eq. (2.10b) since "Yop formally has singularities whenever E = ISn (later, we show that they are not real singularities); thus, we can anticipate that "Yop will, in fact, undergo rapid variations (i.e., resonant behavior) in the vicinity of the 'singularities.' To this we should immediately add that Eq. (2.15) can have a discrete spectrum providing ISn < Il l' where IS 1 is the energy of the first excited state of the target. [That is because only in that region is P in Eq. (2.l3a) such that limri -+ OO
P'I' = limri -+ oo 'I' and limri-+ooQ'I' = O. The latter implies that if QHQ has eigenvalues ISn( < Ill)' they are discrete, and the corresponding eigenfunctions are quadratically integrable:
The true spectrum of QHQ above III is continuous; but if we diagonalize QHQ with a quadratically integrable basis, we will obtain discrete eigenvalues even above Ill' but the apparent singularities there do not correspond to resonances.
The second item concerning "Yop we would like to add is that even for ISn < Ill' the apparent singularities E = ISn in Eq. (2.l0b) are not real singulari ties. To see this, let "Yop operate on the true solution 'I' of the Schrodinger equation
Using Eq. (2.6) in the form
<WnQHP'I') = <wnQ(E - H)Q'I') = (E - ISn)< wnQ'I')
on the right-hand side RHS of Eq. (2.16) gives
"Y '1') = ~ PHQwn)(E - ISn)<wnQ'I') op 'i' E - ISn
= IPHQwn) <wnQ'I') n
(2.16)
(2.17)
We see that the apparent singularity has canceled out! What is happening, in fact, is that near ISn the wave function varies rapidly but smoothly, so that the phase shift exhibits typical Breit - Wigner behavior.
Let us continue with this analysis of the optical potential by rewriting the optical-potential Eq. (2.7) using the spectral-decomposition Eq. (2.l0b) of the optical potential. We write Eq. (2.7) in the form
(H' - E)P'I' = _ PHQw.) <.QHP'I') E-IS.
(2.18a)
where
(2.18b)
The contribution to the scattering from the resonant part comes from the RHS of Eq. (2.18a), and H' represents the nonresonant part of the scattering. In Eq. (2.18b), the summation is understood to include an integral over the continuous part of the spectrum.
A formal solution of Eq. (2.18), following Feshbach but for the case of electron-atom scattering, has been given in the classic paper by O'Malley and Geltman. (4) That treatment, however, is itself somewhat formal and, except for our own work, has really never been directly used except for variational solutions of the QHQ problem in Eq. (2.15). We, therefore, believe it will be more instructive in the context of the two-electron system to give a more incisive description of the procedure whereby the solution of Eq. (2.18) is effected and where at the same time formal definitions of the resonant quantities will also very explicitly emerge.
To effect a solution (and this is true of scattering equations in general), we premultiply Eq. (2.18a) by < <Po YLO[meaningJ dr2d01 <po(r2)Y!o(OdJ to get
<<PoYLo[Hpp-EJP'P)+ L <<poYLOPHQcJ)n)<cJ)nQHP'P) n('/") E - iffn
< <Po YLOPHQcJ).) < cJ).QHP'P) (2. 19a)
E - iff.
At this point, we explicitly deal with a partial wave solution 'P L whose P part is given by
(2.20a)
where ± indicates singlet or triplet spin. The resultant integro-differential operator coming from the PHP part of H' is the well-known exchange approximation operator(S)
ft'ex U = < <Po YLOIPHP - EIP'P) (2.21)
The integro-differential equation satisfied by U is, more explicitly,
ft' U + L ffn<r,r')U(rl)rl2 dr l = _ ff.(r,rl)U(rl)rl2dri ex n('/'.) E - iffn E - iff.
(2.19b)
where
(2. 19c)
The integral terms in Eq. (2.19b) are the explicit form of the optical-potential
terms in Eq. (2.19b). They can be simplified even more by defining
v~(r 1) == (c!>0(r2) yLO(ndP HQ<l>; (1, 2» (2.22)
In Eq. (2.22), we have appended the ( ± ) labels to distinguish, for the moment, the spatial parts of singlet and triplet wave functions for which
(2.23)
It then becomes clear that using P\{I from Eq. (2.20a), the factor
(<l>nQHP\{I) = (<l>nQH[U(r1) yLO(n1)c!>0(r2) ± U(r2)YLO(n2)c!>0(rdJ>
=2(v~U) (2.24)
from which the optical-potential Eq. (2.19) can finally be written as
fe.xU + L 2von(r)(vonU) = _ 2vo.(r)(vooU) n(To) E-8n E-8o
and concomitantly the kernel in Eq. (2.19b) becomes
"I/n(r, r') = 2vo,,(r)voir')
(2.19d)
(2.25)
To solve Eq. (2.19), we use a Green's technique based on the left-hand side of Eq. (2.19) set equal to delta function
fe.xG + L f "I/,,(r, r')[G(r, r")]r=r,r,2dr' = _ b(r - r") "(1'8) E - 8n rr"
(2.26)
The G is constructed from a combination of regular (g <) and irregular (g» solutions of the LHS of Eq. (2.26)
f "I/,,(r, r')g(r')r'2 dr' feexg+ L =0
"(1'8) E-8"
r=O
r-+ 00
r-+C() r
(2.27a)
(2.27b)
(2.27c)
Note in particular that the nonresonant phase shift '10 is determined from Eq. (2.27a) andg«O) = 0 and that '10 so determined is used in starting the backward integration implicit in Eq. (2.27a). The complete Green's function
solving Eq. (2.26) is then
k-1 G(r, r') = {g < (r)g> (r') g< (r')g> (r)
r<r' r>r'
And, finally, the complete solution of the (radial) optical-potential Eq. (2.19d) is
f 00 G(r, r')"Y.(r', r") U(r")r"2 dr"r,2 dr' U(r) = Uo(r) + 0 E - tI. (2.29)
The U o(r) in Eq. (2.29) is the solution of the nonresonant optical-potential equation that is identical to g < (r) as previously noted
(2.30)
Substituting U(r) in Eq. (2.29) into Eq. (2.20a) gives for P'P after some manipulation
P'P) = P'P) G"Y.P'P > o + E-8 ' •
(2.20b)
where it is understood that integration in the last term does not involve all variables
(G"YsP'P) = (G"YS[U(rl) yLO(ntl4JO(r2) ± U(r2) yLO(n2)4Jorl])
= < G(r1' r')"Y.(r', r")U(r") > yLO(nd4Jo(r2)
In Eq. (2.20b), we also have used
P'P 0) = U 0(r1) YLO(ntl4JO(r2) ± U 0(r2) yLO(n2)4JO(r1)
To repeat, P'P 0) satisfies
(4Jo YLO[H' - E]P'P 0) = 0
(2.31)
(2.32)
(2.33)
We remind the reader that we are seeking a solution for P'P, Eq. (2.18), which we have converted into Eq. (2.20b). Multiply Eq. (2.20b) by (CI>.QHP; using Eq. (2.22) and the exchange property of CI>.(1,2) = ± CI>.(2, 1) to obtain
2< U) = 2( U) (CI>.QHPG"Y.P'P) Vo. VO. 0 + E _ 8
• (2.34)
The same properties and Eq. (2.25) allow us to simplify the last term in Eq. (2.34). In detail,
<CI>.QHPG"Y.P'P)= 2 (CI>.(r1 , r2)QHP< G(r1, r'dvo.(r~»,\ yLO(n1)4JO(r2»rl,r2
·(vo.(r'DU(r~»'i (2.35)
where the subscripts on the kets are the coordinates over which we are integrating. Substituting Eq. (2.35) into Eq. (2.34) and reverting to our more abbreviated integration notation gives
< U) - < U) <VO.U) <.QHPGVO.YLOCPO) Vo. - Vo. 0 + E _ C
• (2.36a)
< Vo. U) = -:-----:-=----:::-::-::-=-<--=V--=.o=--. U-=-:o=-:)--:----:--:-:-::::-_,-,- 1-< .QHPGvo. YLol/Jo)/(E - C.)
(2.36b)
Equation (2.36b) is the essential part of the desired solution, since substituting into Eq. (2.20b) without the premultiplication by < <1>. yields
G"Y.P'P) G"Y.P'P 0)
E-rl. E -rl.-~. (2.37)
where ~. is the shift, which, from (Eq. 2.36b), is explicitly written as follows:
~. = < .(r1,r2 )QHP< G(rl,r~)vO.(r~) ),~ YLO l/JO(r2 ) )",'2 (2.38)
Finally, substituting Eq. (2.37) into Eq. (2.20b), we obtain our derived and explicit expression for
P'P) = P'P 0) + G"YsP'P 0 )
E -rl.-~. (2.20c)
The asymptotic form of P'P 0) is from Eqs. (2.27b), (2.30), and (2.32)
1. ) _ sin(kr1 - Ln/2 + '10) (n)", ( ) 1m P'Po - k YLO Ul '1'0 r2 ,,-+ 00 r 1
(2.39)
However, from Eq. (2.20c) we can deduce that P'P) has a different normalization (hence, the subscript u) as well as phase shift
I· Ul ) _ sin(kr1 - Ln/2 + '10 + '1,) v (n)", ( ) 1m py u - .fLO Ul '1'0 r2
,,-+00 (cos '1,) krl
(2.40)
(2.41)
(2.42)
We see from Eq. (2.41) that the actual position ofthe resonance (assumed isolated) occurs at
E=E.=C.+~. (2.43)
i.e., E. is shifted from the eigenvalues 8. of QHQ by an amount Ll. given in Eq. (2.38), which now can legitimately be called the shift. Furthermore, the shape of a hypothetical cross section associated with the purely resonant part of the phase shift "r (i.e., sin2"r) is from Eq. (2.41) purely Lorentzian about E = E. with full width r. at half-maximum (which is usually called the half-width) given by Eq. (2.42).
We can express P'I' in terms of the resonant phase shift by using Eq. (2.20c) in Eq. (2.41), and the definition of "f/., Eq. (2. 19c), without premulti plying by < ¢o YLO ,
tan "r P'I'u = P'I'o - -l-GPHQs> (sQHP'I'o> (2.20d) Irs
In order for P'I' to have the usual asymptotic form, we normalize Eq. (2.20d) by multiplying by COS" .. i.e.,
~ [tan"r ] P'I'> = cos '1r P'I' 0> - trs GPHQs> < <l>sQHP'I' 0> (2.44)
We can rewrite this as
P'I' > = cos '1rP'I' 0> - sin '1rP'I' 1 > (2.45)
where, using the definition in Eq. (2.42) of rs in Eq. (2.44), P'I'l can be written as
(2.46)
P'I'l like P'I' 0 is also a slowly varying function of energy E. To obtain the total wave function '1', we have to know Q'I'. It is now hopefully obvious from Eq. (2.6) that we can write
Q'I'> = I-1-n> (nQHP'I' > n E-$n
(2.6')
It is convenient to define Q-space Green's functions in addition to the P space Green's function defined in Eq. (2.28)
(2.47)
(2.48)
(2.6")
Q'P = G~)QHP[cos'1.P'I' 0> - sin'1.P 'I' 1>]
+ E~c9' cD.>[cos'1.<cD.QHP'I'o>-sin'1.<cD.QHP'I'l>] (2.49) •
Let us note the last square bracket can be simplified in terms ofr. and fl.: using Eqs. (2.41) and (2.36b), we find
. sin'1.{E - s.) [cos '1. < cD.QHP'I' 0> - SIO'1. < cD.QHP'I' 1>] = - (r.k/2)1/2 (2.50)
so that
~ _ (.) (.). sin'1.cD.> Q'I'> - GQ QHPcos '1.P'I' 0> - GQ QHP SIO '1.P'I' 1> - (r.k/2)1/2
= cos '1.Q'I' 0> - sin '1.Q'I' 1> (2.49')
Therefore, using 'I' = P'I' + Q'I' we have from Eqs. (2.45) and (2.49')
'I' > = cos '1. '1'0> - sin '1. '1'1 >
where
'1'1 = P'I'l > + Q'I'l >
(2.51)
(2.52)
(2.53)
The P'I'l is given in Eq. (2.46) and Q'I'l can be inferred from Eq. (2.49')
Q'I'l> = ~<'I'o;~QcD.> + G~)QHP'I'l> (2.54)
These 1 expressions will be used in deriving equations for the line shape parameter and radiation absorption profile discussed in the next section. [The preceding derivation is given in abbreviated form in Ref. 26]
We conclude this section with some observations on the nonresonant function '1'0. It is already explicit in Eqs. (2.49') and (2.52) that '1'0 has a Q'I' 0 as well as a P'I' 0 part. This is at first surprising, because '1'0 is often referred to as the "nonresonant" continuum function: the word nonresonant seems by definition to imply that '1'0 is strictly in P space. However, as is clear from the definition in Eq. (2.48) of G~), Q'I' 0 contains the effects of all resonances other than the sth resonance. It is clear that from the point of view of the sth resonance, other resonances are indeed a part of its background. However to group them as part of the nonresonant background represents an alternative approach from the traditional one that is embodied in the Breit - Wigner
formalism and also incorporated in the Fano approach(2) as a generalization of his single-level results. We shall discuss this interesting difference more at the end of the next section; here, we conclude with a presentation of the nonresonant equations satisfied by qt 0 as distinct to the well-known equation satisfied by pqt o.
We assert that the complete nonresonant function
qt 0 = pqt 0 + Qqt 0
satisfies
This latter equation has the eigenspectrum
n = 1,2,3, . .. s -1,s + 1, ...
(2.52')
(2.55)
(2.56)
(2.57)
where the tffn and <l>n are precisely those ofthe original QHQ Eq. (2.15). But from Eq. (2.57), we see that Eq. (2.56) lacks the sth states of QHQ. This is precisely what we desire, because when we construct the optical-potential equation associated with the pqt 0 part of Eq. (2.55)
[PHP + PHQ E _ QHQ + ~.Q.> (.Q QHP - E Jpqto = 0 (2.58)
and expand in terms of the eigenfunctions of Eq. (2.56), we see from Eq. (2.57) that
The RHS of Eq. (2.59) is exactly the nonresonant potential defining pqt 0'
Eq. (2.18). In addition, using the appropriate relationship between pqt 0 and Qqto,
(2.60a)
We see on expanding in terms of Eq. (2.57) that this is identical to the second term in Eq. (2.52)
(2.60b)
13
In addition to their occurrence as resonances in (electron) scattering, autoionization states may also manifest themselves by radiative decay to, or absorption from, a truly bound state. In that case, the line profile of the radiation becomes another characteristic parameter of the states involved, the autoionization state in particular. This is because, as we shall see, the line profile is largely determined by the fact the resonant state can truly autoionize in competition with its radiative decay.
A classic example of autoionization-state line profiles were those measured by Madden and Codling(6) in vacuum UV photoabsorption of noble gases. A corresponding classic theoretical description of such line profiles was given by Fano(2), where, for an isolated resonance, he showed that the ratio of resonant to nonresonant radiative cross sections can be written as
I<'PEI TI'Pg)12 _ (e + q)2 I <t/lEI TI'Pg)12 - 1 + e2
where t/I 9 is the ground state, e is a dimensionless energy
E-E. e=--
!r.
(3.1)
(3.2)
and q is a ratio of radiative transition matrix elements of an electromagnetic transition operator T between resonant and nonresonant functions. For the purposes of this section, we can let T be the dipole length operator
(3.3)
where Zi is the Z coordinate of the ith electron, and here we take i = 1,2. The quantity q can have either sign, so that from Eq. (3.1), we see that the radiative line will be enhanced on one side and diminished on the other side of the resonance at E = E.; q is accordingly called the line profile parameter.
One of our purposes in this section is to derive an explicit expression for q in terms of the Feshbach formalism. We shall find that the expression is somewhat different from that given by Fano.(2) The difference is due to a subtle difference in the mode of description inherent in the Feshbach formalism from that used by Fano or, more precisely, in the underlying Breit-Wigner approach?) to which we shall return at the end of this section. First, let us derive q.
To do this, we note that the total wave functions labelled 'PE and t/lE by Fano are clearly 'P and 'Po, respectively. Hence, using Eqs. (2.51) and (2.52) for
the above in the LHS of Eq. (3.1), we find that
/ <'PITI'Pg) /2 . 2 <'PoITI'Pg) =lcos'1,-sIn'1,ql (3.4)
where the desired q is
(3.5a)
and 'P 1, 'Po are given by Eqs. (2.52) and (2.53), respectively. That q has been defined consistently with Eq. (3.1) can readily be checked by using Eq. (2.41) in dimensionless form [cf. Eq. (3.2)]
tan '1, = - l/e (2.41')
and noting then that the RHS of Eq. (3.4) does, indeed, reduce to the RHS of Eq. (3.1)!
Substitution for 'P 1 in Eq. (3.5) gives our final expression for q, which is essentially that used for calculations
We write
15
(3.6)
As we have already indicated the resonance parameters, as they come out of the Feshbach theory, are energy dependent, very weakly energy dependent to be sure, but for the purpose of precision calculations, this variation can be important. However, from the point of experiment, determining resonance parameters usually comes from fitting the data to resonance expressions while assuming energy-independent parameters. Furthermore, there is considerable motivation for having energy-independent parameters from the point of view of traditional resonance theory, where resonances are identified as poles in the complex energy plane whose positions are by definition energy independent. In fact, from such a theoretical background, there has now come, in the form of complex rotation, developments that allow this approach to be a useful tool for resonance calculations (cf. Chapter 4 by Junker and Chapter 5 by McCurdy in this volume).
However, in this chapter we shall strictly confine ourselves to the pragmatic aspects whereby results from a Feshbach calculation are directly compared with an experimental fit based on energy-independent resonance parameters. We shall call the latter the Breit-Wigner parameters. To derive this relationship, we start with the equation for the resonant phase shift in the Feshbach form
-1[ tnE) ] '1(E) = '1o(E) + tan tff + L\(E) _ E (4.1)
The Feshbach resonance energy, which we here denote as EF , is defined as the solution to the equation
(4.2)
But note that L\(EF) as such does not appear in Eq. (4.1). Let us nevertheless expand around EF
nE) = r F + r~~E L\(E) = L\F + L\~~E
where
(4.4b)
(4.4c)
(4.4d)
(4.4e)
The first derivative of '1(E) from Eq. (4.1) can then be written
Using Eq. (4.6) in Eq. (4.1) gives
'1(E) = '1o(E) + tan -1 [!r F/(l - Ll~)J + O(bEf EF-E
(4.5)
(4.6)
(4.7)
Observe that all the parameters appearing in Eq. (4.7) are now independent of E, and therefore, can now be directly compared with the Breit-Wigner form, i.e.,
tan = tan -1[ !rBW ] -l[!rF/(I-Ll~)J EBW-E EF-E
from which we derive(8)
Esw = EF + second-order correction
(4.8)
(4.9)
(4.10a)
It is emphasized that these relationships are only of first order, as stated in Eq. (4.1 Oa). In Eq. (4.9), the first-order correction is explicitly given; the apparent absence of a first-order correction in Eq. (4.10a)is due to the fact that EF itself already contains a first-order correction Ll(EF) [cf. Eq. (4.2)]. Nevertheless, the question ofthe second-order correction for EBW is of interest. Treating the resonance as an isolated pole in the complex energy plane,
Drachman(9) has derived the result
(4.tob)
Drachman has also pointed out,(9) and now we may verify, that EBW as given in Eq. (4. lOb) is the energy at which d217r/dE2 = 0, where 17r is the resonant (second) term in Eq. (4.5). The shift from EF is due to the asymmetry ofthe tan - I around its singularity at E = EF • The fact that the resonance does not occur at the singularity of tan 17r comes about as follows: if one defines EBW = EF , then the resultant nonresonant phase shift 170 will not be a smooth function of E, as Drachman has confirmed for us in terms of the example given in his paper.9 However, from a practical point of view, since the correction in Eq. (4.9) to r F is first order, this correction is much more important than the correction in Eq. (4.lOb) to EF (but not CF), which is second order. We shall show a specific example of that in our precision result for H-(2s2; IS).
5. V ARIA TIONAL CALCULATION OF Cn
In this section, we discuss the calculation of the resonance position Cn.(IO) From the point of view of the Feshbach theory, the calculation of Cn
constitutes the heart of the resonance problem. For if Cn and its associated eigenfunction Qn are known accurately, then all other resonance parameters and corrections can be calculated from it effectively by perturbation theory. And in addition, Cn and <l>n can be calculated from the Rayleigh-Ritz variational principle, and thus very accurately b[CnJ = ° where
[C J = <QLSHQLS> = < <l>LsQHQLS > n < QLSQLS > < <l>LSQLS >
(5.1)
since Q2 = Q. Our chief contribution to this field has been the introduction and actual
calculation of Cn using Hylleraas expansions for the <l>LS. The use of Hylleraas coordinates with projection operators requires a nontrivial integration problem that we solved as outlined in Ref. 10, but which we shall treat in a more general way here. For doing actual calculations, we have also consistently used the symmetric Euler angle decomposition of the two electron fixed-nucleus problemY I) The reader is referred to our review paper(1l) for details; we shall repeat here only the most salient aspects. Equation (5.1) is stationary with respect to variations in the wave function. This gives rise to a set of secular equations that can be solved for the eigenvalues for each set of nonlinear parameters. The eigenvalues are minimized with respect to the nonlinear parameters in the wave functions <l>LS. The <l>Ls is the spatial part of the wave function of angular momentum L, parity
( - 1)\ and spin S of the system, where S = 0 for singlet states and S = 1 for triplet states. The <l>LS can be written as(Z4)
_ ,"" { (Ke 1Z ) sJ K+ <l>LS - ~ cos -2- [fLK + ( - 1) LKJEeL
+ sin (K~lZ ) [fLK - ( - 1)s lLKJEeL- }
<l>LS(r 1 , r 2 ) = (-l)sLS(r2 , rd
(S.2a)
(S.2b)
where EeL + are the exchange rotational harmonics(11) whose arguments are our symmetric Euler angles, which are usually written e, qy, tjJY 1) The ILK are functions ofthe residual internal coordinates that in one way or another define the triangle formed by electrons 1 and 2 and the nucleus. For Hylleraas expansions, the ILK are
(S.3a)
and by definition
lLK(r l' r z, r 1Z) == ILK(rz, r l' r 12) (S.3b)
which are such that <l>LS(r1, rz) has the correct symmetry under exchange (r1 ~rz)·
In Eq. (S.3), the angular momentum Lis assumed to be of the same parity as K[i.e., (- I)L = (- Itl U( - I)L+ 1 = (-1)\ then L is replaced by L+ 1 in Eq. (S.3a). The double prime on the sum in Eq. (S.2a) indicates that only K'S of a given parity are included; also K :::::; L. The elmn are the linear parameters and ex.
and f3 are the nonlinear parameters. We can simplify(lO) the expectation value in Eq. (S.1). Using Eq. (2.13a),
we write
z < <l>LsQHQLS) = < <l>LsHLS) - I [< <l>LSPiHLS) + < <l>LsHPi<l>LS) J
i= 1
2 Z
+ I I < <l>LSPiHPj<l>LS) + <LsHP 1PZ<l>LS) i=l j=l
z - I [<LSP1PzHPi<l>LS) + < <l>LSPiHP1PZ<l>LS) J
i= 1
(S.4)
Given the (anti) symmetry of the <l>LS' Eq. (S.2b), it can be seen that the
<rf>LSPiPjHPjrf>LS) = <rf>LSPiPjHPirf>LS)
< rf>LSPiHP jrf>LS) = < rf>LSP jHPirf>LS)
Using these equations in Eq. (5.4) gives
19
(5.5a)
(5.5 b)
<rf>LsQHQrf>LS) = <rf>LsHrf>LS) - 2[ < rf>LSP1 Hrf>LS) + < rf>LsHP1 rf>LS) ]
+ 2< rf>LSP 1HP 1 rf>LS) + 2< rf>LSP 1 HP2rf>LS )
+ <rf>LsHP1P2rf>LS) + <rf>LSP1P2Hrf>LS)
- 2[ <rf>LsP1P2HP1rf>LS) + <rf>LSP1HP1P2rf>LS)]
+ <rf>LSP1P2HP1P2rf>LS) (5.6)
The two quantities in each of the preceding square brackets are the transpose of each others, i.e., for any terms i) and j) in the expansion of rf>LS), we have for the matrix elements
(5.7)
There are three basic kinds of terms in Eq. (5.6). The first is < rf>LsHrf>LS), which occurs in ordinary Hylleraas-type variational calculations of bound states, this term need not be discussed. The second is when P 1 P 2 occurs but not an individual Pi; an example is <rf>LSHP1P2rf>LS)' To demonstrate the calculation of this term, let us write rf>LS in symbolic form
(5.2c)
where P represents the Euler anglesY 1) Using the definition of the projection operators PI and P 2' we find
P 1 P 2rf>LS = </>o(r d</>0(r2) J dr 1 dr2</>0(r 1)</>0(r2)fdr l' r2, r 12 )!?}L(JJ) (5.8)
The volume element can be written asll
dr1dr2 = r1dr1r2dr2r12dr12dp
we can reduce Eq. (5.8) to the form
P 1 P 2rf>LS = c5LO(8n2)1/2</>0(r 1)</>0(r2)J dr</>o(r d</>0(r2)!L(r1, r2' r12) (5.12)
We see that this term is zero unless L= 0, since the </>o(r) is independent of angles. The remaining integral in Eq. (5.10) can be carried out easily. Thus, the
integral <Ls HP 1P2 <1>LS) can be readily obtained and is only nonzero for S states (L= 0).
The third type of terms containing only one Pi is more difficult, and the calculation of such terms was the major technical accomplishment that allowed us to initiate these types of calculations.(10) Let us enlarge the definition in Eq. (2.11) of projectors to include the angles
PI = 4>o(r d YLO(Q2) < 4>o(r 1) YLO(Q2)
and
(5.13a)
(5.13b)
It can be verified that with the definition of the projection operators, the basic properties, such as Eq. (2.3), remain unchanged, PI <l>LS is therefore
(5.14)
The YLO(Q2) can be expanded(ll) in terms of flh(fJ) functions (only L is involved, because both are eigenfunctions of e with eigenvalues L(L + 1). Suppressing summation over other indices, we can write
YLO(Q2) = iX.L((J12)f!)dfJ)
Here, e12 is the angle between r 1 and '2' Let us note
d'l dn2 = ridr1dn 1dn2 = ridr1 sin (J12d(J12dP
_ 2d r12dr I2d'R - r 1 r 1 II
r1r2 r 1
= -drlr12dr12dp r 2
Substituting Eqs. (5.15) and (5.16) into Eq. (5.14), we obtain
4>o(r 1) yLO(n2) II P1<l>LS = r2 rldrlr12dr12iX.L(eI2)fdrl,r2,r12)
(5.15)
(5.16)
(5.17)
For calculational purposes, it is convenient to redefine thefL functions
(5.18)
In that way, the integral in Eq. (5.17) can always be reduced to the form ofthe LHS of the following equation:
I I r 1 drl r12dr 12 cos e12gL(r1,r2, r12)
= f frldrlr12dr12(ri +2~:~rf2 )gL(r1,r2,r12 ) (5.19)
The RHS of Eq. (5.19) can straightforwardly be integrated if the function gL(r 1, r2 ,r12)is of conventional Hylleraas form. Calling this result F(r2 )/r2 , we have
(5.20)
Therefore, referring back to Eq. (5.6), we can infer that
(5.22)
Once again, this integral is of the same form as that occurring in bound-state calculations. Other terms involving Pi can be similarly evaluated.
Relevant precision calculations of Sn will be given in the following sections.
6. PRECISION CALCULATION OF THE LOWEST 1S RESONANCE IN ELECTRON-HYDROGEN SCATTERING
The lowest 1S resonance in electron-hydrogen (e-H) scattering is the most basic resonance in atomic physics. The first (complete) calculation ofthis state, which was a close coupling treatment by Burke and Schey,(13) has been followed by a large number of subsequent calculations. It is not our intention to review these calculations, including even our own, except for the one calculation(8) that we believe deserves the appellation of precision.
Since we can very accurately evaluate S using our Hylleraas expansion described in previous sections, what is required is a very accurate evaluation of A [cf. Eq. (2.43)]. However, A is not easily directly evaluated; thus, the key to the calculation is the observation by Chung and Chen,<14) which avoids having to calculate A directly. In fact, we are now in a position to derive very quickly the basic relationship of Chung and Chen,04) From our foregoing derivations, particularly in Eq. (2.6), we can infer that
< Q'I' > = < <1>. QHP 'l'u > • U E -So
(6.1)
Substitute for P'P u in Eq. (2.20d) to obtain
< ct>sQ'Pu > =_1_[<ct>sQHP'Po> E-Cs
_ t~n'7r < ct>sQHPGPHQct>s > <ct>sQHP'Po >] Irs (6.2)
Using the fact the integrals on the RHS are related to the width or shift, we rewrite the RHS as
1 (r )1/2( Il) <ct>sQ'Pu>=E_Cs 2k 1+ E _ s
Es (6.3 a)
= (rs)1/2_1 2k E-Es
(6.3 c)
Observe that the second term on the right is a function of E and all the quantities can be calculated in precision fashion. The resonance position Es is obtained at the energy where the second term is zero. In order to determine all the quantities in Eq. (6.3 c), C and ct>, precisely (henceforth dropping the subscript s), Ho et al.(8) obtained the minimized C from a Rayleigh-Ritz variational principle by using a Hylleraas-type trial function given in Eq. (5.2a). Specifically, for a lS state, it is given by
ct>(r1,r2 ) = e- arl -(lr2 "L C'mnr~riri2 + (1+-+2) (6.4) l,m,n
(This problem was discussed in Section 5.) The authors found that the minimum value of C is obtained when IX = p, an expected result, since the lowest resonance is dominated by configurations 2S2 + 2p2. The results of minimizing QHQ for various numbers of terms are shown in Table 1. They correspond to the number of terms such that
1+m+n=w (6.5)
where w = 4, 5, 6, 7, and 8. The corresponding number of terms are N = 22, 34, 50,70, and 95 in the expansion in Eq. (6.4). We obtained the lowest eigenvalue for N = 95 when IX = P = 0.49, and this has converged to six significant figures.
To solve for P'P, the optical potential in Eq. (2.16)
l' ="LPHQct>n><ct>nQHP op n E - Cn
(6.6)
TABLE 1 Optimization o{the Lowest H-eS) Eigenvalue ofQHQa
N
0.39 - 0.29748096 - 0.29752739 - 0.29755590 - 0.29756211 - 0.29756491 0.40 - 0.29748709 - 0.29753143 - 0.29755779 - 0.29756301 - 0.29756536 0.41 - 0.29747690 - 0.29753635 - 0.29756032 - 0.29756429 - 0.29756603 0.435 - 0.29744880 - 0.29753725 - 0.29756125 - 0.29756490 - 0.29756638 0.45 - 0.29739988 - 0.29753509 - 0.29756132 - 0.29756527 - 0.29756663 0.46 - 0.29735282 - 0.29753132 - 0.29756081 - 0.29756539 - 0.29756675 0.47 - 0.29729131 - 0.29752502 - 0.29755973 - 0.29756539 - 0.29756684 0.48 - 0.29721227 - 0.29751539 - 0.29755796 - 0.29756525 - 0.29756689 0.49 -0.29711215 -0.29750145 - 0.29755529 - 0.29756494 - 0.29756689 0.50 - 0.29698708 -0.29748199 - 0.29755146 - 0.29756440 - 0.29756685
"Results are in Ry.
was constructed by Ho et al.(14) from the spectrum of eigenvalues and eigenfunctions obtained by calculating QHQ. Because the electrons are indistinguishable, we must normalize P'l' as follows
1 P'l' = j2[u/(rd ¢o(rz) + U/(r 2 )¢0(rd] (6.7)
The optical-potential equation thus reduces to the integro differential equation for u/(r)
[:r22 - 1(1 ~ 1) + k2 + VSI }/(r) + f [Vex(r,r') + Vop(r, r')]u/(r')dr' = 0 (6.8)
where U/(r) = (1/r)u/(r) YIO(Q). The static and exchange potentials VSI and Vex constitute the well-known
(static) exchange approximation. They are given in many places, starting with Morse and Allis.(5) The optical-potential Vop in Eq. (6.8) is obtained by premultiplying Eq. (6.6) by < ¢o YIO to obtain
f ( ') (')d'- ~ <¢ol/oPHQcI>n><cI>nQHP'l'> Vop r,r u/ r r - r L... n=l E-tffn
(6.9)
The equation for the nonresonant continuum P'l' 0 IS obtained by omitting the n = s = 1 term Vop; using the form
(s) _ 1 - _ P'l'o - j2[u/(rd¢0(r2 )+u/(r2 )¢0(rd] (6.10)
Thus, we obtain an analogous equation for ii(l = 0)
[::2 + P + Vst(r) ]ii(r) + f [Vex(r,r') + Vop(r,r')]ii(r')dr' = 0 (6.11)
where
fv- ( ')-( ')d ' = ~ < cPOPHQn > < <l>nQHPlJIo > op r,r u r r r L- E D
nets) - co n (6.12)
Equations (6.8) and (6.11) were solved by converting them to integral equations and numerically solving the resulting equations.(S,1S) This is done by using the Green's function associated with the plane-wave part of those equations. The Green's function is, in analogy to Eq. (2.28) (r>or < are the greater and lesser of rand r')
g(r,r') = -~sin kr < cos kr>
so that the integral equation for the solution of Eq. (6.8) is
u(r) = sin kr + f g(r, r')U(r, r')u(r")dr'dr"
(6.13)
(6.14)
where U is the symbolic sum of the potentials Vst, Vex> and Vop. A similar equation is obtained for ii(r) by replacing Vopt by v.,p in U on the RHS of Eq. (6.14)
fi(r) = sin kr + f g(r, r')U(r, r')fi(r")dr'dr" (6.15)
As radial counterparts of the functions PlJI 0> and PlJI u > of Eqs. (2.39) and (2.40), the functions fi(r) and u(r) are normalized as
lim ii(r) = sin(kr + "0) (6.16a) ,-> 00
I. () sin(kr+,,) Imur =---- '->00 COS",
(6.16b)
Here, the nonresonant and total phase shifts "0 and" actually come from solutions of the respective integral in Eqs. (6.14) and (6.15). The resonant phase ", in Eq. (6.16b) is
", = " -"0 (6.17)
Ho et al.(S) obtained u(r) and ii(r) at N' = 48 Gaussian integration points by solving a system of approximating algebraic equations. They used 16 Gauss-Legendre points for the range from the origin to r = Rand 32 Gauss Laguerre points to cover the remaining range R to 111.75 + R. The results were checked by changing R from 3.5 to 4.5 and 5.5. From the solutions of u(r)
and l1(r) for each E, P'I' and P'I' 0 can be calculated as well as the shift Ll as a function of E.
Referring to Eq. (6.3c), we plot
= E - [r(E)j2k] = $ Ll(E) Yl < <l>Q'I' > + (6.18a)
Y2=E (6.18b)
as a function of E. In Fig. 1, these curves are shown with the nonlinear parameters r:t = P = 0.49. The intersection of Yl(E) and Y2(E) defines the resonant energy Er • It is noted that the specific values of Yl (E) form a straight line to high precision and Er is obtained to the desired accuracy. From Er and $, Ll(Er) is
Ll(Er) == Er - $ (6.19)
However, it should be stressed that Ll(Er) is essentially only of theoretical interest, but by knowing E" we have from Yl the Feshbach width
r(Er) = r F
-.2970 -.2960 EIRyl
FIGURE 1. Graphical determination of IS resonance energy in e-H scattering [cf. text and Ref. 8].
TABLE 2 Convergence Behavior of Resonant Parameters and Test of Numerical Accuracy·
N 34(w=5) 50 (w = 6) 70(w=7) 95 (w=8)
$ 9.5580730 9.5573405 9.5572093 9.5571826 9.5571731b 9.5571758'
N'=40 E, 9.5582611 9.5575476 9.5573679 9.5573617 ~ + 0.0001881 +0.0002071 +0.0001586 +0.0001791
R=4.5 r 0.0476415 0.0470261 0.0470590 0.0470605
N'=48 E, 9.5582636 9.5575501 9.5573703 9.5573642 ~ +0.0001906 +0.0002096 +0.0001610 +0.0001816
R=3.5 r 0.0476482 0.0470327 0.0470656 0.0470670
N'=48 E, 9.5582615 0.5575480 0.5573683 9.5573621 ~ + 0.0001885 +0.0002076 +0.0001590 +0.0001795
R=4.5 r 0.0476418 0.0470263 0.0470593 0.0470608
N'=48 E, 9.5582580 9.5575446 9.5573648 9.5573587 ~ +0.0001850 +0.0002041 +0.0001555 +0.0001704
R=5.5 r 0.0476327 0.0470173 0.0470502 0.0470517
"Results are based on a = {3 = 0·49. See text for definitions of N' and R. Units are in eV (1 Ry = 13.605826eV).
"From Dw - DW - 1 = emP•
("From bw - Jw - 1 = c'aW.
Results taken from our previous calculation are presented in Tables 2 and 3. The reader is referred to our paper(8) for a detailed discussion of extrapolations and other tests of the accuracy of the results. Suffice it here to restate that we believe that the calculation presently represents the most accurate calculation of a resonance. We would like only to reemphasize the point that this calculation also includes the correction between Feshbach and Breit - Wigner resonance parameters as discussed at the end of Section 4. Specifically, we find
aadl = 0.002328 eV E E=Er
(6.20)
Thus, noting from Table 2 that r F(Er ) = 0.0470605, we find from Eq. (4.9)
row = 0.04717 ± 0.00002eV
The second-order correction to d, Eq. (4.l0b), however, as implicit in Section 4, is of higher order in Er if Er » r, which generally is the case. Thus, we find that
EBW = EF = 9.55735 ± 0.00005 eV
TABLE 3 IS Phase Shifts and Widths in the Vicinity of the IS Resonancea
E k '1(E) '1o(E) ll(E) r(E) (eV) (a.u.) (rad) (rad) (eV) (eV)
9.529972 0.836919 1.56723 0.85618 1.162(-4) 0.0470794 9.532693 0.837038 1.61929 0.85621 1.225( - 4) 0.0470776 9.535415 0.837158 1.67771 0.85625 1.287( - 4) 0.0470757 9.538136 0.837277 1.74323 0.85629 l.350( - 4) 0.0470739 9.540857 0.837397 1.81658 0.85633 1.413( -4) 0.0470720 9.543578 0.837516 1.89829 0.85636 1.476( -4) 0.0470702 9.546299 0.837635 1.98861 0.85640 1.538( - 4) 0.0470683 9.549020 0.837755 2.08726 0.85644 1.601 ( -4) 0.0470665 9.551742* 0.837874 2.19329 0.85648 1.664( -4) 0.0470646 9.562626* 0.838351 2.64701 0.85665 1.918( - 4) 0.0470572 9.565347 0.838471 2.75394 0.85669 1.981(-4) 0.0470553 9.568069 0.838590 2.85368 0.85673 2.045( -4) 0.0470534 9.570790 0.838709 2.94519 0.85678 2.109( - 4) 0.0470515 9.573511 0.838828 3.02808 0.85682 2.172(-4) 0.0470496 9.576232 0.838948 3.10256 0.85686 2.236( - 4) 0.0470477 9.578953 0.839067 3.16913 0.85691 2.301(-4) 0.0470458 9.581674 0.839186 3.22848 0.85695 2.365( - 4) 0.0470439 9.584396 0.839305 3.28139 0.85700 2.429( -4) 0.0470420 9.587117 0.839424 3.32860 0.85705 2.493( -4) 0.0470401
'Mesh is 0.0002 Ry from tIf except those marked *, which are for 0.0004 Ry. The width in this table is the Feshbach width, which is to be distinguished from the Breit - Wigner width. See, for example, Table
4 and discussion. These values of E correspond to the points in Fig. I (a = fJ = 0.49; N = 95 terms).
TABLE 4 Lowest IS resonance of H- a
Authorb Reference Method Er(eV)
Burke and Taylor 12 Scattering 9.5603 Shimamura 18 Kohn variational 9.5574 Bardsley and Junker 16 Complex rotation 9.5572 Bhatia 21 Stabilization 9.5570 Doolen et al.' 15 Complex rotation 9.5570 Bhatia and Temkin 7 QHQ and polarized 9.5564
orbital ±0.OO2 Chung and Chen 5 Uncorre1ated 9.5569
Kohn-Feshbach Present work This calculation 9.55735
±0.OOOO5 (C = 9.557175 ± 0.2 x 10- 5)
(ll = + 0.00018 ± 0.5 x 10- 4 )
Experiment Sanche and Burrow 20 9.558 ±O.O1O Williamsd 9.557 ± 0.01
'Results are in eV (I Ry = 13.605826eV; see Ref. 24). "Reference numbers are those given in Ref. 8. 'G. D. Doolan, J. Nuttal, and R. W. Stagat, Phys. Rev. A 10, 1612 (1974). dJ. F. Williams, J. Phys. B 1, L56 (1974).
(eV)
The results compare excellently with the most accurate experiments(16,17) and well serve as a standard of comparison for other calculations (Table 4).
7. REVIEW OF SEMIPRECISION CALCULATIONS OF OTHER FUNDAMENTAL AUTOIONIZATION STATES OF He AND H-
We have previously discussed the significance of the 1 P autoionization states of He**e P) as fundamental checks of the continuum solutions of the Schrodinger equation. In this section, we discuss our calculation of these plus D and S states of He and H - . Although these calculations are not quite at the level of rigor and precision of the calculation of the 1 S resonance of H discussed in the Section 6, they still represent a very high level of accuracy that has very recently been strikingly confirmed by a new measurement of the lowest and most fundamental He**(2s2p) autoionization state.28 Specifically, we shall see that this new measurement confirms a whole series of calculations that had persistently deviated from a previous measurement whose central value had long been believed canonical, without sufficient attention having been paid to the concomitant experimental error.
In this connection, let us note at the outset that a radiative-absorption profile described by the RHS of Eq. (3.1) has its maximum absorption at e = q - 1, which can be derived by simple differentiation. This corresponds to
r. Emaxabs = E. + 2q (7.1)
a fact that must be included when accurate comparison with experiment is made of E •.
The results to be described were all based on Hylleraas variational calculations of<l>L,s and yielded tf to generally high precision. However, for the nonresonant continuum, '1'0 or P'I' 0, as the case may be, we have used various approximations, which we discuss later, in integral expressions for the small quantities r., Eq. (2.42); !1., Eq. (2.38); and q, Eq. (3.5a) et seq.
The nonresonant continua we have used correspond to three increasingly elaborate approximations of P'I' 0 of Eq. (2.32): (1) the exchange approximation(5)
'I' 0 -+ 'I'ex = fi[ U~d ~0(Od</>0(r2) ± (1 +-+2) ] (7.2)
(2) the polarized orbital approximation(18,19)
'1'0 -+ 'I'g'0) = ,~{ U~1) [</>0(r2) + </>(pol)(r l' r2)] Y/O(Od ± (1 +-+2) } (7.3)
where ¢(POI) _ 8(r1' r2) u1S -+ p(r2) cos 912
- - d r 2 In and
(7.4)
(7.5)
(Z is the nuclear charge); (3) the (nonresonant) polarized pseudo-state approximation(20)
tTl tT/(pSS) 1 [F1(r 1 ) y (n)"" ( ) To-t TO = M -- 10 U1 0/18 r 2
y2 r 1
+ L F1(r2) «P2p(r2)YlI.1 (;\,r2) + (1 +-+2)J 1=0,2 r 2
(7.6a)
'P\rSS) has been written explicitly for total P waves, which is the only case to which it has been applied; «P2P is defined and discussed below. The angular factor YlI,l in Eq. (7.6a) represents two ways a p state can be coupled to a p-scattered wave to form a total P wave
YlI,l == L(llm - milO) Y1,m(!l1) Y1, -m(!l2) (7.6b) m
The justifications of these approximations should, at this point be clear; however, we shall discuss them briefly. The exchange approximation in Eq. (7.2) is clearly the first nontrivial approximation; it satisfies the equation
(PHP - E)'P·x = 0 (7.7a)
and it is automatically in P space
P'P.X = 'P.X (7.8)
The only nonobvious fact about this approximation is that has a discrete as well as a continuous spectrum. (20) In fact, for charged targets (Z ;;:: 2), the discrete spectrum is (convergingly) infinite; the discrete functions satisfy
(PHP - E~X)'P~X = 0, E~x < - Z2 (7.7b)
The 'P~.X) are of the form in Eq. (7.2) where, however, the functions u become the exponentially damped function, u -t uv(r), which are orthonormal
(7.9)
These discrete terms contribute, for example, to the shift; if we make a spectral expansion of the Green's function, then(20)
L\ =~{L rv + .9fqEI)d~'} 2n v Es - Ev E - E
=Ab+Ac (7.10)
where f!IJ represents a principal value integral and r. are the discrete counterparts of the width function in Eq. (2.42)
(2.42')
In all our calculations where a discrete contribution to L\ has been included, the exchange approximation for r. has been used irrespective of what approximation has been used for r(E').
As compared to the exact equation for P'P 0, which in a slightly more symbolic form than Eq. (2.19a) satisfies
[ PHP + L PHQcI>n> <cI>nQHP - EJP'Po = 0 (7.11) n(rS) E - tin
the exchange approximation in Eq. (7.7a) lacks the nonresonant-optical potential terms
"f'"(S) = L PHQcI>n><cI>nQHP n(rS) E - tin
(7.12)
The additional approximations we have considered attempt in an increasingly incisive manner to incorporate the effect of "f'"(s). The polarized orbital approximation(18.19) identifies this contribution with induced polarization and incorporates it via Eq. (7.3) in well-known ways. The nonresonant polarized pseudo state is adapted from the approximation of Burke et al.(21), which is itself a variant of the closely coupled polarized orbital approximation of Damburg and Karule.(22) We have discussed that variation in a previous article;(23) suffice it here to state that the nonresonant character of Eq. (7.6) comes from identifying the resonant state with the fiJ2p, fiJ2s states of the target and taking qJ2P as a polarized (orbital) pseudostate that is orthogonal to them. This is done by letting
qJ2 (r) = u1S_p(r) p r (7.13)
where
uls_p(r) = au1S_p(r) + bR2p(r) (7.14)
and a and b are such that < uls_p (r)R 2P(r» = O. With regard to polarized orbital and polarized pseudo state approximations, the functions 'PlfO) and 'PljSS) are what come out of the calculation, but P'P( =F 'P) is obtained by the operation of P, Eq. (2.13b), on them, and P'P is what is used in calculating the specific-resonant quantities.
We shall not go into detail about the numerical solutions nor the reduction of the formulas to actual calculational form, since that has been discussed in our previous papers.(20.24) Rather, we shall give a compilation of
TABLE 5 3 P and 1 D Resonances of H-
State
Our calculations 'D
8(Ry) -0.2851969 -0.256174 8(eV)a 9.72549 10.12037 Nonresonant approx. Pol. orb. Pol. orb. L\,(eV) 0.0130 0.0040 L\b L\ = L\, + L\b 0.0130 0.0040 E(eV) 9.7385 10.1244 r(eV) 0.0063 0.010
Other results E,p r,p E'D r'D
Scattering Burke, Ref. 13 9.7417 0.0059 10.1267 0.0088
Stabilizationb 9.7403 0.0049 Sanche and Burrow, 9.738 0.0056 10.128 Ref. 6 ±0.01O ±0.0005 ±0.01O Williams' 9.735 0.0060
±0.0005
aResult relative to ground state of H using R = 13.605826eV. bA. K. Bhatia, Phys. Rev. A9, 9 (1974); 10, 729 (1974). '1. F. Williams, J. Phys. B 7, L56 (1974).
0.0073 ±0.002
31
our final results for the states mentioned at the beginning of this section, taking the opportunity to correct various typographical and other errors in one or two of those entries.
In Table 5, we give our results for two higher e P, 1 D) resonances of H-(e - H). The nonresonant continuum used then was the polarized orbital approximation. Note that H- has no bound-state spectrum in these symmetries, so that the shift has only a continuum part. The agreement with the two experiments cited,06.17) which are generally conceded to be the most accurate experiments, is very satisfactory.
Coming now to helium, we show in Table 6 our results for the 1.3S, 3p, and 1,3 D states.(24) The nonresonant continua used are as noted in footnote c, and the agreement with experiment [cf. in particular the compilation of Martin(25)] are in all cases within the experimental error, which is quoted by Martin as being ofthe order ± 0.05 eV. It is only important to note that to the extent these states, being dipole forbidden from the ground state, are observed through electron impact, it is appropriate to use the rydberg of infinite mass
TABLE 6 Summary of First Four Autoionization States of He ofS, P, and D Angular Momentum
(Excluding 1 P )
A 8 l (Ry) 8(eV)" r(eV) L\,(eV) A.(eV) A(eV) E",,(eV)
's 1 -1.5576265 57.8223 1.25 x 10-' 1.177 X 10- 2 9.346 X 10- 3 2.112 X 10- 2 57.8435 2 -1.2454971 62.0691 6.67 x 10- 3 5.307 X 10- 3 1.664 X 10- 2 2.195 X 10- 2 62.0911 3 -1.1801597 62.9581 3.87 x 10- 2 3.355 X 10- 3 9.959 X 10-" 4.351 X 10- 3 62.9624 4 -1.0964678 64.0968 2.412 x 10- 3 1.478 X 10- 3 2.748 X 10- 3 4.226 X 10- 3 64.1010
3S 1 -1.2052107 62.6173 4.21 x 10- 5 4.73 X 10- 8 2.061 X 10- 5 2.066 X 10- 5 62.6173 2 -1.1195332 63.7830 7.09 x 10-" 1.97 X 10- 5 8.294 X 10- 5 1.026 X 10-" 63.7831 3 -1.0976990 64.0800 1.22 x 10-" 6.60 X 10- 5 2.353 X 10-" 3.013 X 10-" 64.0803 4 -1.0650259 64.5246 1.32 x 10-" 3.24 x 10-" 1.594 X 10-" 1.626 X 10-" 64.5247
3p 1 -1.522983 58.2937 8.90 x 10- 3 1.455 X 10- 2 1.267 X 10-2 2.720 X 10- 2 58.3209 2 -1.169776" 63.0994 2.61 x 10- 3 5.013 X 10- 3 2.219 X 10- 3 7.232 X 10- 2 63.1066 3 -1.158012" 63.2594 4.88 x 10- 5 4.730 x 10-" 6.825 X 10- 5 7.298 X 10- 5 63.2595
'D 1 -1.405634 59.8903 0.0729 0.0220' 0.0023' 0.0243 59.9146 2 -1.138752 63.5215 0.0187 0.0044 0.65 x 10-" 0.0045 63.5259 3 -1.112855 63.8738 5.81 x 10-" 1.815 X 10-" 0.10 X 10-" 1.915 X 10-" 63.8740 4 -1.073241 64.4128 7.12 x 10- 3 1.667 X 10- 3 0.56 X 10- 5 1.673 X 10- 3 64.4145
3D 1 -1.167611 63.1288 2.72 x 106 , 2.60 x 10-4 , 9.91 X 10-" 1.25 X 10- 3 63.1301 2 -1.121361 63.7581 1.92 x 10-" 9.17 X 10- 5 3.62 X 10- 6 9.53 X 10- 5 63.7582 3 -1.083336 64.2755 3.31 x 10-6 1.20 X 10-" 2.78 x 10-' 3.98 x 10-' 64.2759 4 -1.066831 64.5000 1.36 x 10-' 5.02 X 10- 5 5.50 X 10- 6 5.57 X 10- 5 64.5001
"Results in eV relative to the ground state oCR. - 5.80744875 Ry [CO L. Pekcris,Phys. Rev. 146,48 (1966)] using R = 13.605826eV Crom B. N. Taylor, W. R. Parker, D. N. Langenberg, Rev. Mod. Phys. 41, 375 (1969).
'From Chen and Chung, ReC. 14. 'I, 3, D widths and A" have heen calculated with polarized orbital nonresonant Cunctions. All other results use the exchange approximation.
(RaJ to effect the conversion to electron volts. Only for photo-excitation experiments should RM be used; this has been shown in the appendix of Ref. 24.
We now come to the final set of calculations that we wish to discuss here-the 1 P autoionization states of He (Table 7). Here, we also calculated q in view of the fact that the experiment, being one of photoabsorption, also determines q. Our most recent calculation(26) has now included all the different contributions (bound, continuum, and other resonant) to q that we discussed in Section 3. These contributions are small for the wide resonances (s = 1,3), but note the relative size of C;qb for the narrow resonance (s = 2). All these calculations used the exchange approximation for the nonresonant con tinuum after we were convinced of the reliability of that approximation. This was done by comparing the calculation of A with one involving the nonresonant polarized pseudostate continuum.(20) The value of A given in Table 7 uses the latter but differs from the exchange approximation value by
TABLE 7 Final Results for Resonance Parameters
He
Parameter s=1 s=2 s=3
qo - 3.139 -3.556 -2.849 (jq, 3.677( -I) 1.678 3.635( -I) (jqb -7.631( -2) -2.636 -1.129( -I) (jq, -9.892( -4) -9.387( -2) -4.595( -3) q -2.849 -4.606 -2.604 E(eV)a 60.1450 62.7594 63.6610 r(eV)a 0.0363 1.06 ( -4) 0.009 ~(eV)a -0.00729
Ref. 27 q -2.80 ±0.25 -2.0± 1.0 E(eV) 60.l33±0.015 62.756 ± 0.010 63.656 ± 0.007 r(eV) 0.038 ± 0.004 0.008 ± 0.004
Ref. 28 q - 2.55 ± 0.16 -2.5±0.5 E(eV) 60.151 ± 0.010 63.655 ± 0.01 r(eV) 0.038 ± 0.002 0.0083 ± 0.002
aResults for E and r for He from our previous paper [Ref. 24].
less than 0.15%. That may be anomalously small, but the corresponding widths differ by about 1.5%, which is probably a more realistic estimate ofthe accuracy of the exchange approximation.
The reason we went to the trouble (with the valuable collaboration of Prof. Burke) of using the pseudostate approximation(20) for Ll was the persistent deviation of our previous results for the position E, from the central value of the classic experiment of Madden and Codling.(27) However, as we have indicated, the calculated value has persisted. We are, therefore, very gratified that in a recent experiment by Morgan and Ederer,(28) done at the National Bureau of Standards in a division headed by Dr. Madden, a new value of E, has been found that is completely in accord with our calculations of that quantity over the years. While trying to avoid any hubris and emphasizing that the assessment of the comparative inherent experimental accuracy is not within our purview, we must nevertheless conclude that the agreement with the recent experiment does, indeed, provide strong support for the essential correctness and cogency of the Feshbach formalism, which it has been the purpose of this chapter to elucidate.
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to. A. K. Bhatia, A. Temkin, and J. F. Perkins, Phys. Rev. 153, 177 (1967). 11. A. K. Bhatia and A. Temkin, Rev. Mod. Phys. 36, to50 (1964). 12. P. G. Burke and H. M. Schey, Phys. Rev. 126, 149 (1962). 13. P. G. Burke, in Invited Papers of the Fifth International Conference on the Physics of
Electronic and Atomic Collisions 1967 edited by P. M. Branscomb, p. 128, Univ. of Colorado, Boulder 1968.
14. K. T. Chung and J. C. Y. Chen, Phys. Rev. A 13, 1655 (1976). 15. Y. K. Ho and P. A. Fraser, 1. Phys. B 9, 3213 (1976). 16. L. Sanche and P. D. Burrow, Phys. Rev. Lett. 29, 1639 (1972). 17. J. F. Williams, 1. Phys. B 7, L 56 (1974); Electron and Photon Interactions with Atoms (edited by
H. Kleinpoppen and M. R. C. McDowell, p.309, Plenum, New York 1976). 18. A. Temkin, Phys. Rev. 107, 1004 (1957). 19. A. Temkin and J. C. Lamkin, Phys. Rev. 121, 788 (1961). 20. A. K. Bhatia, P. G. Burke, and A. Temkin, Phys. Rev. A 8, 21 (1973); 10, 459 (1974). 21. P. G. Burke, D. F. Gallaher, and S. Geltman, 1. Phys. B 2, 1142 (1969). 22. R. J. Damburg and E. Karule, Proc. Phys. Soc. (London) 90 637 (1967). 23. R. J. Drachman and A. Temkin, in Case Studies in Atomic Collision Physics, vol. 2 (edited by E.
McDaniel and M. R. C. McDowell, p.399, North-Holland, Amsterdam 1972). 24. A. K. Bhatia and A. Temkin, Phys. Rev. A 11,2018 (1975). 25. W. C. Martin, 1. of Phys. and Chen. Ref Data 2, 257 (1973). 26. A. K. Bhatia and A. Temkin, Phys. Rev. A 29, 1875 (1984). 27. R. P. Madden and K. Codling, Astrophys. 1. 141, 364 (1965). 28. H. D. Morgan and D. Ederer, Phys. Rev. A 29, 1901 (1984).
CHAPTER TWO
TARGETS
1. INTRODUCTION
In Chapter 1, we dealt with the theory and applications of the projection operator formalism to two-electron systems. The reason for singling out two electron autoionization systems (implying a one-electron target) was given in Chapter 1; to repeat briefly, the fact that the target eigenfunctions are known analytically and the special character of a one-electron target allow exact projection operators in the Feshbach senseI!) to be written down, which thereby allows precision calculations of the composite (two-electron) system approaching the accuracy of ordinary two-electron bound-state systems to be carried out [cf., for example, Ref. 2]. In the case of autoionization of two- or more electron target systems, the inability of achieving comparable precision arises from two causes: not only can we not give the exact eigenfunctions analytically, but we cannot even supply a formal, explicit expression for the projection operators that we need at the outset.
The problem of writing down a projection operator even formally arises because of the requirement of antisymmetry: the operator must preserve the indistinguishability of scattered and orbital electrons. It might be thought that we could simply symmetrize the obvious asymmetric operator, but such a procedure would destroy idempotency, which is a necessary property in the original Feshbach theory (we shall see later, however, that it can be dispensed with to some extent).
A. TEMKIN and A. K. BHATIA • Atomic Physics Office, Laboratory for Astronomy and Solar Physics, Goddard Space Flight Center, National Aeronautics and Space Administration, Greenbelt, MD 20771.
35
Feshbach(1) also dealt with the problem of anti symmetry; unfortunately, his formal solution to the problem was incomplete or, more accurately, heuristi cally intended, in that he did not include spin coordinates at all, and angular momentum considerations are at best implicit. In addition, the final form of his projection operators contains a partial antisymmetrizer on both sides of a projector, which is very confusing until we note that the operator itself can be put into a manifestly symmetric form.
One object then of this chapter is to put Feshbach's derivation and notation into a complete form, so that we can directly construct and apply the projection operator in an actual calculation. Of course, to do so will involve approximating the eigenfunction of the target and, having done that, solving for the eigensolutions of an ensuing homogeneous equation, in terms of whose solutions the projection operator is expressed. The practicality of this proce dure will be demonstrated for the open-shell approximation of the He like targets. This, hopefully, will provide not only a valuable pedagogical example of how the construction works in practice but also a useful (and first) idempotent operator of the simplest nonseparable approximation of the He like (ground-state) target (Section 2).
We shall also derive explicitly the kernel of the auxiliary integral equation using a Hylleraas-type of target-state wave function. Although we have not been able to solve the eigenvalue equation analytically, we can readily write down a variational principle for the eigensolutions. Since they are one-dimensional functions, such variational (or numerical) solutions may be more convenient to work with for calculational purposes.
In Section 3, we shall drop the question of idempotency and review the ideas associated with quasi-projection operators (QPO). This review will be brief, since most of the basic material has already been covered in the literature. One interesting sidelight is the fact that our new notation for projection operators (above) leads naturally to the same form of QPOs that we had originally introduced quite independently of Feshbach's theory.
In the case of QPOs, the formalism can be explicitly generalized to treat autoionization states that lie in the region of inelastic scattering, and this is also reviewed. Special attention is given to a recent calculation of the lowest 2 pO resonance of He -. This state is of interest because it was thought to be a shape resonance. We shall, therefore, take this opportunity to discuss the difference between these kinds of resonances from a novel point of view (in terms of the literature; we have, in fact, discussed this for many years). In the case of the above 2 pO resonance, although Feshbach (i.e., core excited) in nature, it is a special type that can occur only in many-electron targets. It is suggested that some other resonances, heretofore considered shape re sonances, may be of this special Feshbach type.
From the spectrum of these quasi-projection or projection-operator
PROJECTION AND QUASI-PROJECTION OPERATORS 37
calculations, we can readily construct a (quasi-) optical potential that we would expect to lead to a convergent method of calculating nonresonant phase shifts. In practice, this scheme (and various modifications) have never worked out. We conclude this chapter with a preliminary explanation of this phenomenon (Section 4).
2. IDEMPOTENT RESONANCE PROJECTION OPERATIONS
2.1. Notation and Preliminaries
We start by considering the scattering of an electron from an atomic target (atom or ion) consisting of N electrons. The total (time-independent) wave function 'P has the asymptotic form
lim 'P = ( - 1 )Pi sin (kri - nl/2 + '11) 1/1 o(r(i». ~~oo k~
(2.1.1)
The 'P is a fully antisymmetric function of the coordinates of all i = 1,2, ... , N + 1 electrons
'P = 'P(x1 , ••• , Xi> ••. , Xi'.··' XN+l) = - 'P(x, ... , Xi'···' Xi'···' XN+l)'
where Xi are the totality of coordinates (spin and space) of the ith electron
(2.1.2)
A cyclic permutation of the particle labels i that brings Xi into the first position is denoted by Pi
( 1,2, ... ,
(2.1.3)
[This implies that ( _1)Pi is the parity of Pi; in particular, ( - I)P' = 1 always; cf. below.] The channel functions 1/10 in Eq. (2.1.1) is defined as the target func tion ¢o coupled to the orbital angular and spin of the scattered particle
I/Io(r(i» = L (liLomiMoILMdHSom.MsoISMs)Ylimi(ni)X~hm.¢o(x(i» mag q.n.
(2.1.4)
Rotation invariance and spin independence ofthe Hamiltonian will guarantee that the physical results we shall be concerned with here are independent of the total magnetic quantum numbers ML and Ms. In general, we shall take them to be the most convenient values commensurate with the magnetic quantum numbers of the target Mo and Mso.
In Eq

Autoionization: Recent Developments and Applications

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